Register No:………………… Name :…………............ M.TECH. DEGREE EXAMINATION
Model Question Paper - II Second Semester Branch: Information Technology Specialization: Network Engineering MITNE 205.4 ADVANCED TOPICS IN GRAPH THEORY (Elective III) (Regular – 2011 Admission onwards) Time: Three hours Maximum: 100 Marks 1. a)Define planar graph and plane graph. Give examples. Prove that n − e + f = k + 1 , where n is the number of vertices, e is number of edges , f is the number of faces and k is the number of components of a plane graph G . (10 marks) b)The complete graph of 5 vertices is nonplanar.
(15 marks)
OR 2. a)Define geometric dual and combinatorial of a given graph G. Prove that a graph hadual iff it is planar
(15 marks)
b) Let G be a simple graph with less than 30 edges. Prove that G has a vertex v such that d(v)≤4
(5 marks)
c) Show that the complete graph of 4 vertices is self dual 3. a)Define directed graph.Explain different types of diagraph with examples
(5 marks) (5 marks)
b)Prove that ,a graph G is Orientale iff it is connected and has no bridges.
(15 marks)
c)Give an example of 2-regular diagraph with 5 vertices
(5 marks)
OR 4. a)Prove that every edge in a diagraph belongs either to a connected circuit or a directed cut set (10 marks) b)Explain Hopcraft and Tarjan algorithm. Illustrate the algorithm, using suitable examples, for Produce a strongly connected orientation (15 marks) P.T.O.
5. a) State and prove Menger’s theorem
(15 marks)
b) Prove that maximum flow possible between two vertices a and b in a network is equal to the minimum of the capacitor of all cut-sets w.r.to a and b
(10 marks)
OR 6 a) Let f be a flow of network N=(V,A)and let f have value d. If A( X , X ) is a cut in N then d = f ( X , X ) − f ( X ,X ) and d ≤ c( X , X ) .
(15 marks)
b) Every circuit has an even number of edges is common with any cut set
(10 marks)
7. Prove that for any graph G a) χ (G) ≤ ∆(G) + 1
(10 marks)
b) χ (G ) = 2 iff G is bipartite
(8 marks)
c) χ (G ) ≥ 3 iff G has an odd cycle
(7 marks)
OR 8. a) Let G be a nonempty bipartite graph then χ 1 (G ) = ∆(G ) b) A map G is 2 face colorable iff it is an Euler graph
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